Thanks to Shatner's suggestion — and the ensuing support from fans and Trekkies — Vulcan is currently in the lead by a healthy margin, followed by Cerberus and Styx. David Dickinson tells us more:

This astronomical horse-race has the propensity to get interesting. In order to be considered, the IAU's naming convention simply states "Those that share Pluto's orbital rhythm take the name of underworld deities," And the named moons of Charon, Nix & Hydra all follow this convention. Shatner's case for Vulcan does cite the god as "The nephew of Pluto" in Roman mythology, but anyone who had studied Roman and Greek mythos knows that familial relations can be proven between nearly any given god and/or goddess.

Interestingly, [researcher and co-discoverer Mark] Showalter turned down Shatner's second Star Trek/mythological suggestion of Romulus, citing that Romulus and Remus are already the names of the moons of asteroid 87 Silvia. While the "double naming" of objects in the solar system isn't unheard of, it may be a definite strike against a proposal. Cerberus, Orpheus, Hypnos & Persephone are all names in the running that are all also assigned to asteroids.

There's more at Universe Today, including precedents for 'Vulcan' as a name for celestial bodies.